I, like many of you I suspect, take copious notes when reading and working through math exercises/theorems/constructions. I have stacks and stacks of notes ranging from one day old to several years. It often turns out that something I'm working through is actually something that I have taken notes on previously but since there may have been several intervening years I don't actually recall taking the notes and, even if I did, my notes are in such a disheveled state that I would have little chance of actually finding anything specific. Many times, I have attempted to go through and organize them but end up with little piles everywhere and finally throw up my hands in frustration and toss them back in the box.

I am not looking for a technical solution to this problem as discussed here. What I seek is a mathematical information model of sorts that can overlay a structure on the knowledge that I acquire and annotate. If such a model exists, it would probably come in the form of a directed dependency graph with arbitrarily deep nesting. Any problem, theorem, example, counterexample, definition should be able to fit naturally in some node (although could possibly exist in more than one node). The problem here though is any such model is bound to be extremely complicated and a very practical problem is how to label the nodes in a concise but meaningful way.

I'm not looking for a way to organize "all of mathematics" but a natural way to organize the mathematics that I learn. Finally, within this context, I have the following questions:

Can anyone suggest an effective means by which this organization can be accomplished? Has anyone attempted to organize their notes/thoughts via the directed graph idea and, if so, what were the results?

2 Answers
2

I haven't used the directed graph idea. At least, not explicitly. However, you may find the following comments useful.

I try to organize my notes in the following way. I have separate LaTeX files (which I compile into searchable PDF files) corresponding to different math subjects, e.g., Statistics, Combinatorics, Optimization, etc. Within each document, I have sections corresponding to different concepts. For example, in Statistics, I have Random Variable, Probability Distribution, etc. Sections can be ordered alphabetically and/or logically (for example, Statistics goes like this: Probability Space, Random Variable, Mathematical Expectation, Variance, Probability Distribution, etc.) Each section starts with the definition of the concept, followed by discussion of the definition, examples, counter-examples, notes, small problems, references, good books or papers, Internet links, "see also" notes, etc. Sections may have subsections, like for example Probability Distribution may have subsections corresponding to different popular distributions (Gaussian, Log-normal, Poisson, etc.) If I feel that some note in, say, Combinatorics, is useful in Statistics, I often will copy and paste this note into Statistics (that is, documents corresponding to different math subjects may have duplicate bits of notes). I think that this scheme is pretty general, flexible and convenient, and may be used in many circumstances.

After many iterations, I believe I have developed a successful system for organizing the notes that I take and the math that I learn. The solution is actually a combination of technology and methodology and is somewhat similar to the approach suggested by Leo in another answer. I am writing all of this out in the hopes that it will be beneficial to other learners.

First, I have come up with rather broad categories into which I can categorize what I'm learning. There is nothing particularly orginal about these categories and they are rather standard; just a way for me to bucket thigs. It's a tree structure that breaks down along rather obvious lines. For example, my top categories include Algebra, Analysis, Geometry, Topology. Each of these categories breaks down further. Exactly how is breakdown occurs is under flux; I'm trying not to go beyond 3 or 4 levels in order to keep things reasonable. If I am working through a large amount of a particular text, I will create a category for the text itself and all notes on the text, irrespective of their mathematical classification, will be placed in this category.

Now, with regard to the actual notes, there are three basic levels and the amount of effort grows with the levels. First, when I am reading somewhat randomly and am not going line-by-line through a text, I have an Excel file that I record the topic, author, source and other pertinent details about what I'm reading. The amount of information is no more than what can fit on a row of a spreadsheet and doesn't require mathematical symbols. I typically make entries here for things that I want to look at in detail later.

The next level involves actually writing things down including: worked problems, proved theorems, definitions, etc. I do this legibly with pencil and paper and then scan these pages in batches. Once these are scanned, I organize them in a folder according to the structure I described earlier. At the time of this writing, I currently have about 600 pages of these notes.

At the final level, I texify them. This, obviously, is the most laborious part but it can also be very benefical. Before I tex something up, I read over it carefully and make sure that everything is in order because once I actualy start typing it's mostly mechanical. At this point, I often find points that I missed or overlooked and it gives me another chance to soak in the material. At the time of this writing, I currently have about 150 pages of these sorts of notes (obviously, I'm quite behind in the texification process).

With regard to the actual tex development environment, I have settled on Texelipse. The notes that are organized according to mathematical subject classificaitons are compiled into a single document and the tex files themselves are organized in folders according the subject classification. The text-specific notes each have their own project/organizational structure that is determined by the text on which the notes are based. As the material in these notes becomes more familiar it will be integrated into the main set of notes.

So, that's a pretty good summary of my approach and so far it seems to be working for me pretty well. The hardest part will be catching up with all the TEX I must write. Putting it into TEX though is very valuable because not only does it force me to review the material but it gives me a nice, clean an organized document that I can refer back to that has been written in my own words.